Fluctuation near jamming

When compressing athermal particles interacting with purely repulsive and short-range potential, the particles suddenly start interacting with each other at a certain density φ_J, causing the system to behave like a solid. This phenomenon is known as the jamming transition, and φ_J is referred to as the jamming transition point. The minimal model for studying the jamming transition is a system of frictionless spherical particles. For this model, the critical behavior of physical quantities such as energy, pressure, and contact number in the vicinity of the transition point has been studied extensively. Based on the systematic numerical simulations, it is now widely believed that the upper critical dimension d_u of the jamming transition is du = 2. However, the physical mechanism that determines this value has not yet been clarified. According to the Ginzburg criterion for the equilibrium phase transition, the upper critical dimension is determined by comparing the mean and variance of the order parameter. Therefore, the first step to discuss the upper critical dimension is to investigate the behavior of fluctuations of physical quantities.

In this work, we perform systematic numerical simulations of sample-to-sample fluctuations of frictionless spherical particles in two dimensions. We consider two different protocols: one with density control and the other with pressure control. We show that for the average value of the physical quantity, the results of the two protocols are in almost exact agreement. On the other hand, for fluctuations, qualitative differences are found between the two results. When pressure is used as a control parameter, the fluctuations do not diverge even near the transition point, whereas when density is used as a control parameter, the fluctuations diverge toward the transition point. In the latter case, we also peform the finite size scaling and derived the correlation length. Surprisingly, we show that the correlation length associated with the fluctuations diverges much faster than that of the mean values.